I discuss the relationship between generalized Farey symbols, finite index sub-groups of $SL(2,\mathbb{Z})$ and their generators, fundamental domains of these groups as special polygons. This is based on the work of R.S. Kulkarni. I also found the series of posts by Lieven Le Bruyn on his blog as a useful starting place.

References

R.S. Kulkarni, An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group, Amer. J. of Math., 113 (1991) 1053-1133.